File size: 4,054 Bytes
d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.MeasureTheory.Falconer.Estimates.DistanceOccupiedSum
import OAI.MeasureTheory.Falconer.Estimates.DistanceWeightSum
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ComplexConjugate
theorem distance_graph_pointwise_bound
(Q₁ Q₂ : DyadicSquare) (ρ ν : Measure Plane) (A B : CellPath) (N a n : ℕ)
(sig t : ℝ) (hA : A.length = a) (hB : B.length = a)
(I : Finset GraphNonedgeIndex)
(hI : I ⊆ sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B)
{θ : ℝ} (hθ : 0 < θ) (hscale : θ = graphUncappedAngularScale N sig t)
(axis d : ℝ) (hd : 0 ≤ d)
(hdegree : ∀ j k s reverse row,
((activeDistanceMatrixNeighbors I (a+n) j k θ axis s reverse row).card : ℝ) ≤ d)
(U V : CellPath → UnitCircle → ℂ) (u w : UnitCircle) :
‖∑ e ∈ graphSupportedDistanceIndices I θ axis,
graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2 (u,w) *
(U e.1.1.1 u*conj (V e.1.1.2 u))*(V e.1.2.1 w*conj (U e.1.2.2 w))‖ ≤
d/2 * ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n,
(‖U row.1 u‖^2*‖V row.2 w‖^2 + ‖V row.2 u‖^2*‖U row.1 w‖^2) := by
classical
let C := sourceProductVertices Q₁ Q₂ ρ ν A B n
let D := sourceProductVertices Q₂ Q₁ ν ρ B A n
let H := graphSupportedDistanceIndices I θ axis
let F := fun e : GraphNonedgeIndex => graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2 (u,w) *
(U e.1.1.1 u*conj (V e.1.1.2 u))*(V e.1.2.1 w*conj (U e.1.2.2 w))
let S := d/2 * ((∑ row ∈ C, ‖U row.1 u‖^2*‖V row.2 w‖^2) +
(∑ col ∈ D, ‖V col.1 u‖^2*‖U col.2 w‖^2))
have hS : 0 ≤ S := by dsimp [S]; positivity
have hh : H ⊆ sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B :=
(Finset.filter_subset _ _).trans hI
have hmatrix (j b : ℕ) :
‖∑ row ∈ C, ∑ col ∈ D, if distanceMatrixIndex row col j b true ∈ H
then F (distanceMatrixIndex row col j b true) else 0‖ ≤
‖graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w)‖ * S := by
have he := distance_matrix_complex_bound_on I (a+n) j b θ axis true C D
(sourceProductVertices_subset Q₁ Q₂ ρ ν A B a n hA hB)
(sourceProductVertices_subset Q₂ Q₁ ν ρ B A a n hB hA)
(fun row => U row.1 u*V row.2 w) (fun col => V col.1 u*U col.2 w)
(graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w)) d (hdegree j b true)
have hf (row col : CellPath × CellPath) : F (distanceMatrixIndex row col j b true) =
graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w) *
((U row.1 u*V row.2 w)*conj (V col.1 u*U col.2 w)) := by
dsimp [F,distanceMatrixIndex]
rw [map_mul]
ring
simpa only [hf,norm_mul,mul_pow,H,S] using he
change ‖∑ e ∈ H, F e‖ ≤ _
rw [source_distance_graph_sum_occupied Q₁ Q₂ ρ ν A B N a n sig t hA hB H hh F,← hscale]
calc
_ ≤ ∑ j ∈ Finset.range (angularBinCount θ), ∑ b ∈ Finset.range (angularBinCount θ),
‖∑ row ∈ C, ∑ col ∈ D, if distanceMatrixIndex row col j b true ∈ H
then F (distanceMatrixIndex row col j b true) else 0‖ :=
(norm_sum_le _ _).trans (Finset.sum_le_sum (fun _ _ => norm_sum_le _ _))
_ ≤ ∑ j ∈ Finset.range (angularBinCount θ), ∑ b ∈ Finset.range (angularBinCount θ),
‖graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w)‖*S :=
Finset.sum_le_sum (fun j _ => Finset.sum_le_sum (fun b _ => hmatrix j b))
_ ≤ S := by
simp only [← Finset.sum_mul]
exact (mul_le_mul_of_nonneg_right
(graphRotatedDistanceWeight_norm_total_le (angularBinCount_pos hθ) axis (u,w)) hS).trans_eq (one_mul S)
_ = _ := by
dsimp [S,C,D]
rw [sourceProductVertices_swap_sum Q₁ Q₂ ρ ν A B n
(fun c p => ‖V c u‖^2*‖U p w‖^2),Finset.sum_add_distrib]
end PlanarFalconer
end OAI
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